Why did the french revolution end the way it did?

The French Revolution ended the way it did because Napoleon saw an opportunity to rule and took it. During the Revolution, there was no sort of military authority; there was only bloodshed and government corruption among the people put in charge. 

Napoleon Bonaparte staged a coup d’état, abolishing the Directory and appointing himself France’s “first consul.” The event marked the end of the French Revolution and the beginning of the Napoleonic era, in which France would come to dominate much of continental Europe.

Tl;dr: France was weak and the practical "throne" was empty, Napoleon took the shot, and appointed himself leader.